EPSRC Reference: 
EP/L013916/1 
Title: 
Quantum groups and noncommutative geometry 
Principal Investigator: 
Voigt, Dr C 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
School of Mathematics & Statistics 
Organisation: 
University of Glasgow 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 June 2014 
Ends: 
31 May 2016 
Value (£): 
97,299

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
27 Nov 2013

Mathematics Prioritisation Panel Meeting Nov 2013

Announced


Summary on Grant Application Form 
Quantum groups are mathematical objects that describe symmetries in mathematics and physics, including phenomena which are related to fundamental questions about space and time. The theory has connections to a large range of fields in mathematics, including representation theory, combinatorics, and operator algebras. Quite remarkably, quantum groups can also be used to study problems in lowdimensional topology, like distinguishing knots and finding invariants of 3dimensional manifolds.
In this project we study a range of questions at the current focus of research in the subject. One main aim is to study what happens if one looks at a quantum group from "far away". Technically, this amount to transport ideas from coarse geometry to the realm of quantum groups, and to consider the large scale properties of the latter. To get an idea of what coarse geometry is about one can imagine the set of integers as a subset of the real line. The local structure of the integers is very different from the structure of the real line  the integers are a discrete set, whereas the real line is a continuous and connected space. However, if we "zoom out" the integral points on the line appear to get closer and closer, and an infinitely far observer will not notice any difference between the integers and the real line. On a large scale perspective, both spaces can still be distinguished from a single point  which means that even from "far away" some amount of information about the dimension of spaces is retained.
Apart from this we shall study problems at the intersection of representation theory of quantum groups and operator Ktheory. Classical representation theory of Lie groups is a vast subject, with applications ranging from number theory to physics. For instance, the properties of elementary particles are determined by representations of the Poincar\'e group, the symmetry group of spacetime. If one deforms a classical symmetry group then typically some new and unexpected phenomena show up. We will investigate in particular the structure of principal series representations of deformed semisimple complex Lie groups represented by Drinfeld doubles. This will help to understand the geometry of quantum flag manifolds and the operator Ktheory of classical quantum groups. Roughly speaking, operator Ktheory is an invariant which can be used to extract homological information from a quantum group and to distinguish among quantum groups.
Our methods combine techniques from various fields in mathematics, most notably coarse geometry, operator algebras, and representation theory, but also differential geometry and category theory, and an overall objective of this project is to provide new links between these areas.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.gla.ac.uk 